Theorem neldif 3392

Description: Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)

Assertion

Ref	Expression
neldif	⊢ ((A ∈ B ∧ ¬ A ∈ (B ∖ C)) → A ∈ C)

Proof of Theorem neldif

Step	Hyp	Ref	Expression
1		eldif 3222	. . . 4 ⊢ (A ∈ (B ∖ C) ↔ (A ∈ B ∧ ¬ A ∈ C))
2	1	simplbi2 608	. . 3 ⊢ (A ∈ B → (¬ A ∈ C → A ∈ (B ∖ C)))
3	2	con1d 116	. 2 ⊢ (A ∈ B → (¬ A ∈ (B ∖ C) → A ∈ C))
4	3	imp 418	1 ⊢ ((A ∈ B ∧ ¬ A ∈ (B ∖ C)) → A ∈ C)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∈ wcel 1710   ∖ cdif 3207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by: (None)